When my friend was handed the trophy, I couldn't help but cheer.
Literal
Friend [subject-が] trophy [object-を] was-handed time [topic-とき], I [topic-は] cheer without [could not help-はいられなかった].
~ずにはいられない ('cannot help but do X') uses the classical negative ず + にはいられない ('cannot remain without doing'). It expresses an irresistible urge. 手渡される is the passive of 手渡す ('to hand over'). 喝采 (かっさい) is a formal word for cheering or applause.